Consider yourself corrected.
I believe that the only way the payments stop early is if they get promoted back, or go into administration.
There is no action if they get relegated again, receive a points sanction or (rather annoyingly) breach PSR rules.
As much as I want to see Leicester punished as severely as possible, it just wouldn't seem fair to remove a clubs parachute payments after getting relegated again, even if that would likely be down to poor self-management. You'd still likely have some players on very high wages that you'd be trying to shift while they'd have even lower income to service that in an even lower league. Seems a good way to ensure clubs go out of business.
Now, whether parachute payments should even be a thing, and whether we need other, fairer, ways to safeguard teams relegated from the ridiculousness that is the PL, is a completely different question. But if you get parachute payments in the Championship, you should still get them in L1 (or even L2) as well.